Introduction to euclid’s geometry


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Introduction to euclid’s geometry

  1. 1. Mathematics activity for FA-2
  2. 2. Geometry  Geometry ( geo "earth", metron "measurement") is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space. A mathematician who works in the field of geometry is called a geometer.  Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers.
  3. 3. Euclidean Geometry  Euclidean Geometry is the study of geometry based on definitions, undefined terms (point, line and plane) and the assumptions of the mathematician Euclid (330 BC)  Euclidean Geometry is the study of flat space  Euclid's text Elements was the first systematic discussion of geometry. While many of Euclid's findings had been previously stated by earlier Greek mathematicians, Euclid is credited with developing the first comprehensive deductive system. Euclid's approach to geometry consisted of proving all theorems from a finite number of postulates and axioms.  The concepts in Euclid's geometry remained unchallenged until the early 19th century. At that time, other forms of geometry started to emerge, called non-Euclidean geometries. It was no longer assumed that Euclid's geometry could be used to describe all physical space.
  4. 4. Euclid’s Definitions  The Greek mathematicians of Euclid’s time thought of geometry as an abstract model of the world they lived in. The notions of point, line,plane etc. were derived from what was seen around them.  Euclid summarised these notions as definitions. A few are given below:  A point is that of which there is no part.  A line is a widthless length.  A straight line is one which lies evenly with the points on itself.  The extermities of lines are called points.  A surface is that which has only length and breadth.  The edges of surface are lines.  A plane surface is a surface which lies evenly with straight lines on itself.
  5. 5. Euclid’s Axioms  Axioms are assumptions used throughout mathematics and not specifically linked to geometry.  Here are some of euclid’s axioms:  Axiom 1: Things that are equal to the same thing are also equal to one another (Transitive property of equality).  Axiom 2: If equals are added to equals, then the wholes are equal.  Axiom 3: If equals are subtracted from equals, then the remainders are equal.  Axiom 4: Things that coincide with one another equal one another (Reflexive Property).  Axiom 5: The whole is greater than the part.  Axiom 6: Things which are halves of the same things are equal to one another  Axiom 7: Things which are double of the same things are equal to one another
  6. 6. Euclid’s Postulates  Postulates are assumptions specific to geometry.  Let the following be postulated:  To draw a straight line from any point to any point.  To produce [extend] a finite straight line continuously in a straight line.  To describe a circle with any centre and distance [radius].  That all right angles are equal to one another.  The parallel postulate: That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.
  7. 7. Equivalent Versions of Euclid’s Fifth Postulate (parallel postulate)  To the ancients, the parallel postulate seemed less obvious than the others. It seemed as if the parallel line postulate should have been able to be proven rather than simply accepted as a fact. It is now known that such a proof is impossible.  Euclid himself seems to have considered it as being qualitatively different from the others, as evidenced by the organization of the Elements: the first 28 propositions he presents are those that can be proved without it.  Many alternative axioms can be formulated that have the same logical consequences as the parallel postulate. For example Playfair's axiom states:  For every line l and for every point P lying outside l, there exists a unique line m passing through P and parallel to l  This result can also be stated in the following way:  Two distinct intersecting lines cannot be parallel to the same line
  8. 8. THANK YOU Done by SrihariSanjee v Class 9 E Roll no. : 38